lecture 11.pdf

Year: 2023-2024 Spring Semester Introduction to Operations Research and Decision Support Dr. Amany Magdy Dr. Hussien Shafei Dr. Asmaa Awad  Midterm Exam From Lecture 1 TO Lecture 8 MCQ  5 questions T/F  5 questions...

Year: 2023-2024 Spring Semester Introduction to Operations Research and Decision Support Dr. Amany Magdy Dr. Hussien Shafei Dr. Asmaa Awad  Midterm Exam From Lecture 1 TO Lecture 8 MCQ  5 questions T/F  5 questions Essay  one question (graphical solution) 2 Review We will provide you some texts mentioned in the lectures and provide examples of questions of the two types (True/False) or MCQ. The aim of these models is to train students on how to expect questions based on what they have studied in this course.  Lecture 1 (1 of 3) In practice, OR does not offer a single general technique for solving all mathematical models. Instead, the type and complexity of the mathematical model dictate the nature of the solution method. T/F: Operations Research (OR) offers a single method to solve all mathematical models. (False) MCQ: Operations Research utilizes which of the following approaches for solving mathematical models? (a) A single universal method (b) Different methods depending on model characteristics CORRECT (c) Only graphical techniques (d) All solutions involve manual calculations  Lecture 1 (2 of 3) Problem definition involves delineating the scope of the problem under investigation. This function should be carried out by the entire OR team. T/F: Problem definition requires identifying the limitations under which the decision is made. (True ) Model validity checks whether or not the proposed model does what it purports to do that is, does it adequately predict the behavior of the system under study? T/F: Model validation involves ensuring the model accurately reflects the real system. (True )  Lecture 1 (3 of 3) An important aspect of the model solution phase is sensitivity analysis. It deals with obtaining additional information about the behavior of the optimum solution when the model undergoes some parameter changes. MCQ: During which phase of an OR study is sensitivity analysis most crucial? (a) Model construction (b) Model validation (c) Solution of the model CORRECT (d) Implementation of the solution  Lecture 2 (1 of 3) Objectives of business decisions frequently involve maximizing profit or minimizing costs. T/F: Businesses often aim to maximize profit or minimize costs.(True) Linear programming uses linear algebraic relationships to represent a firm’s decisions, given a business objective, and resource constraints. T/F: Linear programming uses linear equations to model decisions with constraints.(True)  Lecture 2 (2 of 3) Decision variables - mathematical symbols representing levels of activity of a firm. Objective function - a linear mathematical relationship describing an objective of the firm, in terms of decision variables - this function is to be maximized or minimized. T/F: Decision variables represent the level of activity in a company (e.g., production quantity).(True) The objective function represent the level of activity in a company (e.g., production quantity). (False) The objective function is a linear formula representing a company's goal (maximize or minimize).(True) Decision variables are linear formula representing a company's goal (maximize or minimize) (False)  Lecture 2 (3 of 3) A feasible solution does not violate any of the constraints: The feasible solution area is an area on the graph that is bounded by the constraint equations. MCQ: In a graphical solution of a linear programming model, the feasible solution area represents: a) All possible points on the graph. b) The points that violate at least one constraint. c) The region satisfying all the constraints. (Correct Answer) d) The area where the objective function is maximized.  Lecture 3 (1 of 3) Characteristics of the solution: -The solution point will be on the boundary of the feasible solution area and at one of the corners of the boundary where two constraint lines intersect. -The optimal solution point is the last point the objective function touches as it leaves the feasible solution area. T/F: The optimal solution point typically lies on the boundary of the feasible area, at corners where constraints intersect.( True) The optimal solution is not always the last point the objective function touches the feasible area as it leaves. (False)  Lecture 3 (2 of 3) MCQ: In a graphical solution of a linear programming problem, where is the optimal solution typically located? a) Anywhere within the feasible solution area. b) At the center of the feasible solution area. c) On the boundary of the feasible area, often at corner points. (Correct Answer) d) Completely outside the feasible solution area. A slack variable is added to a  constraint (weak inequality) to convert it to an equation. MCQ: What is the basic purpose of a slack variable in a linear programming model? a) To represent an excess resource above a constraint requirement. b) To directly minimize the objective function. c) To convert an inequality constraint into an equality. (Correct Answer) d) To increase the value of the decision variables.  Lecture 3 (3 of 3) A slack variable typically represents an unused resource A slack variable contributes nothing to the objective function value. T/F: A slack variable typically represents unused resources. (True) A slack variable typically represents used resources. (False) A slack variable contributes nothing to the objective function value. (True) A slack variable contributes a value to the objective function value. (False)  Lecture 4 (1 of 3) A surplus variable is subtracted from a  constraint to convert it to an equation (=). A surplus variable represents an excess above a constraint requirement level. A surplus variable contributes nothing to the calculated value of the objective function. T/F: A surplus variable is subtracted from a greater than or equal constraint to convert it to an equation. (True) A surplus variable represents an amount exceeding a constraint requirement. (True) A surplus variable contributes nothing to the calculated value of the objective function. (True) A surplus variable is subtracted from a less than or equal constraint to convert it to an equation. (False) A surplus variable represents unused resources(False) A surplus variable contributes a value to the calculated value of the objective function(False) Lecture 4 (2 of 3) Infeasible problems do not typically occur, but when they do, they are usually a result of errors in defining the problem or in formulating the linear programming model. T/F: Infeasible problems do not typically occur. (True) Infeasible problems are usually occur. (False) Infeasible problems are usually a result of errors in defining the problem or in formulating the linear programming model. (True)  Lecture 4 (3 of 3) An Unbounded Problem In some problems, the feasible solution area formed by the model constraints is not closed. Value of the objective function increases indefinitely without ever reaching a maximum value because it never reaches the boundary of the feasible solution area. MCQ: What does an unbounded solution in a linear programming model signify? a) All constraints are satisfied, and the objective function reaches a maximum value. b) There is no feasible solution that satisfies all the constraints. c) The objective function can increase indefinitely without reaching a maximum. (Correct Answer) d) The solution point lies within the feasible area.  Lecture 5 (1 of 1) When linear programming was first developed in the 1940s, virtually the only way to solve a problem was by using a lengthy manual mathematical solution procedure called the simplex method. The mathematical steps of the simplex method were simply programmed in prewritten software packages designed for the solution of linear programming problems. The ability to solve linear programming problems quickly and cheaply on the computer, regardless of the size of the problem, popularized linear programming and expanded its use by businesses. T/F: The simplex method is a manual mathematical solution procedure for linear programming problems(True) The simplex method were programmed in software packages for linear programming problems. (True) Computer can solve linear programming problems quickly and cheaply regardless of the size of the problem(True) Manual simplex method can solve linear programming problems quickly and cheaply regardless of the size of the problem(False)  Lecture 6 (1 of 1) Sensitivity analysis determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations. The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal. The sensitivity range for a right-hand-side value is the range of values over which the quantity’s value can change without changing the solution variable mix (or variables that do not have zero values), including the slack variables. T/F: Sensitivity analysis determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations. (True) The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point will remain optimal. (True) The sensitivity range for a right-hand-side value is the range of values over which the quantity’s value can change without changing the solution variable mix, including the slack variables. (True) The sensitivity range for a right-hand-side value is the range of values over which the quantity’s value can change with changing the solution variable mix, including the slack variables(False)  Lecture 7 (1 of 2) In the linear programming models formulated and solved in the previous lectures, the implicit assumption was that solutions could be fractional or real numbers (i.e., non- integer). However, non-integer solutions are not always practical. T/F: Non-integer solutions of linear programming models are not always practical. (True) Non-integer solutions of linear programming models are always practical.  Lecture 7 (2 of 2) Total Integer Model: All decision variables required to have integer solution values 0-1 Integer Model: All decision variables required to have integer values of zero or one. Mixed Integer Model: Some of the decision variables (but not all) required to have integer values. If all decision variables required to have integer solution values, the problem is total Integer linear programming (True) If all decision variables required to have integer values of zero or one, the problem is 0-1 Integer Model Integer linear programming (True) If Some (but not all) of the decision variables required to have integer values., the problem is Mixed Integer linear programming(True) If all decision variables required to have integer values of zero or one, the problem is Mixed Integer Model Integer linear programming  Lecture 8 (1 of 2) Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution. A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution. Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution. (True) Rounding non-integer solution values up to the nearest integer value can result in an feasible solution. (False) A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution. (True) A feasible solution is ensured by rounding up non-integer solution values but may result in a more than optimal (sub-optimal) solution. (False)  Lecture 8 (2 of 2) The traditional approach for solving integer programming problems is the branch and bound method. The branch and bound method is based on the principle that the total set of feasible solutions can be partitioned into smaller subsets of solutions. The traditional approach for solving integer programming problems is the branch and bound method. (True) In the branch and bound method the total set of feasible solutions is partitioned into smaller subsets of solutions. (True) Essay (graphical solution) 1 of 5 The student can refer to the three references of the course, then go to graphical solution part of linear programming problem. He can study the examples solved there. He also can try to solve the exercises on this subject. Essay (graphical solution) 2 of 5 Essay (graphical solution) 3 of 5 Essay (graphical solution) 4 of 5 Essay (graphical solution) 5 of 5